*(English)*Zbl 0905.14008

The author extends Viro’s method of glueing polynomials in order to keep singular or critical points in the process. The input for the glueing method is a subdivision $\left\{{{\Delta}}_{i}\right\}$ of a nondegenerate Newton polyhedron and a compatible system of polynomials ${F}_{i}$ with support on the ${{\Delta}}_{i}$. A sufficient condition for the existence of a polynomial with the same singularities as the ${F}_{i}$ is roughly that for each $i$ the equisingular locus in the space of all polynomials is smooth and transversal to the space of polynomials with the given Newton diagram and coinciding with ${F}_{i}$ for all monomials in ${{\Delta}}_{i}$.

As example plane curves with the maximal number of cusps are constructed for degree eight ($\kappa =15$) and nine ($\kappa =20$). Another application is the asymptotically complete solution to the problem of possible collections of critical points of real polynomials in two variables without critical points at infinity.

##### MSC:

14E15 | Global theory and resolution of singularities |

32S15 | Equisingularity (topological and analytic) |

14F45 | Topological properties of algebraic varieties |

14B05 | Singularities (algebraic geometry) |

14H20 | Singularities, local rings |